rules for predicate logic

Rules for Predicate Logic

The system for predicate logic in this courseis quite simple.

We will learn 3 rules which can be used for proofsand also with trees.

Rules for Predicate Logic

The system for predicate logic in this courseis quite simple.

We will learn 3 rules which can be used for proofsand also with trees.

$O Universal Out#O Existential Out

QE Quantifier Exchange

There are no quantifier In rules.

Taking an Instance

$x(Px>Qx)

Taking an Instance

$x(Px>Qx)

Pa>Qa

Taking an Instance

$x(Px>Qx)

Pa>Qa

To take an instance:1. Remove the quantifier.

Px>Qx2. Substitute the same name (a-w) for each occurrence of the variable.

Pa>Qa

Taking an Instance

$x(Px>Qx)

Pa>Qa

To take an instance:1. Remove the quantifier.

Px>Qx2. Substitute the same name (a-w) for each occurrence of the variable.

Pa>Qa

Pa>Qx WRONGPa>Qb WRONGPb>Qb RIGHT

Taking an Instance

$x(Px>Qx)

Pa>Qa

To take an instance:1. Remove the quantifier.

Px>Qx2. Substitute the same name (a-w) for each occurrence of the variable.

Pa>QaThis should be familiar from algebra. x+7=7+x

3+7=7+3 RIGHT3+7=7+4 WRONG

Universal Out

1) Ba A2) Da A

-$x(Dx>-Bx) GOAL

Universal Out

1) Ba A2) Da A3) $x(Dx>-Bx) PA

?&-? ?,? &I -$x(Dx>-Bx) 3-? -I

Universal Out

1) Ba A2) Da A3) $x(Dx>-Bx) PA4) Da>-Ba 3 $O

?&-? ?,? &I -$x(Dx>-Bx) 3-? -I

Universal Out

1) Ba A2) Da A3) $x(Dx>-Bx) PA4) Da>-Ba 3 $O

?&-? ?,? &I -$x(Dx>-Bx) 3-? -I

$O Rule

$xAxAn

Universal Out

1) Ba A2) Da A3) $x(Dx>-Bx) PA4) Da>-Ba 3 $O5) -Ba 4,2 >O ?&-? ?,? &I -$x(Dx>-Bx) 3-? -I

Universal Out

1) Ba A2) Da A3) $x(Dx>-Bx) PA4) Da>-Ba 3 $O5) -Ba 4,2 >O6) Ba&-Ba 1,5 &I7) -$x(Dx>-Bx) 3-6 -I

Universal Out

1) Gj A2) $x(Gx>Rx) A3) $x(Rx>Ux) A

Uj GOAL

Universal Out

1) Gj A2) $x(Gx>Rx) A3) $x(Rx>Ux) A4) Gj>Rj 2 $O

Uj GOAL

Universal Out

1) Gj A2) $x(Gx>Rx) A3) $x(Rx>Ux) A4) Gj>Rj 2 $O5) Rj>Uj 3 $O

Uj GOAL

Universal Out

1) Gj A2) $x(Gx>Rx) A3) $x(Rx>Ux) A4) Gj>Rj 2 $O5) Rj>Uj 3 $O6) Rj 4,1 >O Uj GOAL

Universal Out

1) Gj A2) $x(Gx>Rx) A3) $x(Rx>Ux) A4) Gj>Rj 2 $O5) Rj>Uj 3 $O6) Rj 4,1 >O7) Uj 5,6 >O

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